1. CHAPTER 2 POLYNOMIAL & RATIONAL FUNCTIONS
2. 2.1 COMPLEX NUMBERS
3. Objectives Add & subtract complex numbers
Multiply complex numbers
Divide complex numbers
Perform operations with square roots of negative numbers
4. i = the square root of negative 1 In the real number system, we cant take the square root of negatives, therefore the complex number system was created.
Complex numbers are of the form, a+bi, where a=real part & bi=imaginary part
If b=0, a+bi = a, therefore a real number (thus reals are a subset of complex #)
If a=0, a+bi=bi, therefore an imaginary # (imaginary # are a subset of complex #)
5. Adding & Subtracting Complex # Add real to real, add imaginary to imaginary (same for subtraction)
Example: (6+7i) + (3-2i)
(6+3) + (7i-2i) = 9+5i
When subtracting, DONT FORGET to distribute the negative sign!
(3+2i) (5 i)
(3 5) + (2i (-i)) = -2 + 3i
6. Multiplying complex # Treat as a binomial x binomial, BUT what is i*i? Its -1!! Why??
Lets consider i raised to the following powers:
7. EXAMPLE
8. Dividing Complex # It is not standard to have a complex # in a denominator. To eliminate it, multiply be a well-chosen one: ( conjugate/conjugate)
The conjugate of a+bi=a-bi
We use the following fact:
9. EXAMPLE
10. 2.2 Quadratic Functions Objectives
Recognize characteristics of parabolas
Graph parabolas
Determine a quadratic functions minimum or maximum value.
Solve problems involving a quadratic functions minimum or maximum value.
11. Quadratic functions, f(x)= graph to be a parabola. The vertex of the parabolas is at (h,k) and a describes the steepness and direction of the parabola given
12. Minimum (or maximum) function value for a quadratic occurs at the vertex. If equation is not in standard form, you may have to complete the square to determine the point (h,k). If parabola opens up, f(x) has a min., if it opens down, f(x) has a max.
This parabola opens up with a steepness of 2 and the minimum is at (1,1). (graph on next page)
13. Graph of
14. 2.3 Polynomial Functions & Their Graphs Objectives
Identify polynomial functions.
Recognize characteristics of graphs of polynomials.
Determine end behavior.
Use factoring to find zeros of polynomials.
Identify zeros & their multiplicities.
Use Intermediate Value Theorem.
Understand relationship between degree & turning points.
Graph polynomial functions.
15. The highest degree in the polynomial is the degree of the polynomial.
The leading coefficient is the coefficient of the highest degreed term.
Even-degreed polynomials have both ends opening up or opening down.
Odd-degreed polynomials open up on one end and down on the other end.
WHY? (plug in large values for x and see!!)
16. Zeros of polynomials When f(x) crosses the x-axis.
How can you find them?
Let f(x)=0 and solve.
Graph f(x) and see where it crosses the x-axis.
What if f(x) just touches the x-axis, doesnt cross it, then turns back up (or down) again?
This indicates f(x) did not change from pos. or neg. (or vice versa), the zero therefore exists from a square term (or some even power). We say this has a multiplicity of 2 (if squared) or 4 (if raised to the 4th power).
17. Intermediate Value Theorem If f(x) is positive (above the x-axis) at some point and f(x) is negative (below the x-axis) at another point, f (x) = 0 (on the x-axis) at some point between those 2 pts.
True for any polynomial.
18. Turning points of a polynomial If a polynomial is of degree n, then it has at most n-1 turning points.
Graph changes direction at a turning point.
19. Graph
20. Graph, state zeros & end behavior END behavior: 3rd degree equation and the leading coefficient is negative, so if x is a negative number such as -1000, f(x) would be the negative of a negative number, which is positive! (f(x) goes UP as you move to the left.) and if x is a large positive number such as 1000, f(x) would be the negative of a large positive number (f(x) goes DOWN as you move to the right.)
ZEROS: x = 0, x = 3 of multiplicity 2
Graph on next page
21. Graph f(x)
22. Which function could possibly coincide with this graph?
